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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,Mod(2,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.19
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.19

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.85418 + 11.7069i) q^{2} +(-46.5389 + 4.59638i) q^{3} +(-11.8772 - 3.85915i) q^{4} +(250.787 - 123.414i) q^{5} +(32.4826 - 553.347i) q^{6} +(-2.39286 - 2.39286i) q^{7} +(-621.573 + 1219.91i) q^{8} +(2144.75 - 427.821i) q^{9} +(979.788 + 3164.76i) q^{10} +(-210.579 - 289.837i) q^{11} +(570.492 + 125.008i) q^{12} +(-15342.4 + 2429.99i) q^{13} +(32.4496 - 23.5760i) q^{14} +(-11104.1 + 6896.28i) q^{15} +(-14422.0 - 10478.2i) q^{16} +(13948.3 + 7107.04i) q^{17} +(1031.69 + 25901.5i) q^{18} +(-21930.5 + 7125.65i) q^{19} +(-3454.92 + 497.997i) q^{20} +(122.360 + 100.363i) q^{21} +(3783.54 - 1927.81i) q^{22} +(-36514.8 - 5783.38i) q^{23} +(23320.2 - 59630.2i) q^{24} +(47662.9 - 61901.3i) q^{25} -184117. i q^{26} +(-97847.8 + 29768.4i) q^{27} +(19.1861 + 37.6549i) q^{28} +(26690.2 - 82143.9i) q^{29} +(-60144.7 - 142781. i) q^{30} +(-1992.92 - 6133.58i) q^{31} +(25487.9 - 25487.9i) q^{32} +(11132.3 + 12520.8i) q^{33} +(-109064. + 150114. i) q^{34} +(-895.409 - 304.784i) q^{35} +(-27124.7 - 3195.56i) q^{36} +(-43852.5 - 276874. i) q^{37} +(-42755.8 - 269949. i) q^{38} +(702849. - 183609. i) q^{39} +(-5328.50 + 382647. i) q^{40} +(14992.3 - 20635.1i) q^{41} +(-1401.81 + 1246.36i) q^{42} +(-132226. + 132226. i) q^{43} +(1382.57 + 4255.12i) q^{44} +(485075. - 371984. i) q^{45} +(135410. - 416750. i) q^{46} +(-305041. - 598678. i) q^{47} +(719346. + 421355. i) q^{48} -823532. i q^{49} +(636293. + 672759. i) q^{50} +(-681808. - 266642. i) q^{51} +(191603. + 30346.9i) q^{52} +(-1.37672e6 + 701472. i) q^{53} +(-167067. - 1.20069e6i) q^{54} +(-88580.5 - 46698.9i) q^{55} +(4406.40 - 1431.73i) q^{56} +(987870. - 432421. i) q^{57} +(912159. + 464768. i) q^{58} +(-1.18678e6 - 862243. i) q^{59} +(158500. - 39056.4i) q^{60} +(-1.74789e6 + 1.26991e6i) q^{61} +(75500.1 - 11958.0i) q^{62} +(-6155.79 - 4108.36i) q^{63} +(-1.09008e6 - 1.50037e6i) q^{64} +(-3.54777e6 + 2.50288e6i) q^{65} +(-167221. + 107109. i) q^{66} +(460041. - 902882. i) q^{67} +(-138241. - 138241. i) q^{68} +(1.72594e6 + 101316. i) q^{69} +(5228.32 - 9917.30i) q^{70} +(3.11520e6 + 1.01219e6i) q^{71} +(-811215. + 2.88231e6i) q^{72} +(573607. - 3.62161e6i) q^{73} +3.32263e6 q^{74} +(-1.93366e6 + 3.09990e6i) q^{75} +287972. q^{76} +(-189.654 + 1197.43i) q^{77} +(846270. + 8.56859e6i) q^{78} +(-5.46749e6 - 1.77650e6i) q^{79} +(-4.91000e6 - 847911. i) q^{80} +(4.41691e6 - 1.83514e6i) q^{81} +(213774. + 213774. i) q^{82} +(-4.03654e6 + 7.92215e6i) q^{83} +(-1065.98 - 1664.23i) q^{84} +(4.37517e6 + 60925.8i) q^{85} +(-1.30278e6 - 1.79313e6i) q^{86} +(-864568. + 3.94557e6i) q^{87} +(484465. - 76731.7i) q^{88} +(3.07550e6 - 2.23448e6i) q^{89} +(3.45535e6 + 6.36842e6i) q^{90} +(42526.8 + 30897.5i) q^{91} +(411376. + 209607. i) q^{92} +(120941. + 276290. i) q^{93} +(7.57424e6 - 2.46102e6i) q^{94} +(-4.62047e6 + 4.49355e6i) q^{95} +(-1.06903e6 + 1.30333e6i) q^{96} +(1.77246e6 - 903112. i) q^{97} +(9.64096e6 + 1.52698e6i) q^{98} +(-575638. - 531537. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 34 q^{3} - 20 q^{4} - 6 q^{6} - 1364 q^{7} - 10 q^{9} + 3160 q^{10} + 2018 q^{12} - 16868 q^{13} - 37570 q^{15} + 507892 q^{16} + 106550 q^{18} + 70800 q^{19} - 6 q^{21} - 195160 q^{22} - 444200 q^{25}+ \cdots - 22089384 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{20}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.85418 + 11.7069i −0.163888 + 1.03475i 0.759395 + 0.650630i \(0.225495\pi\)
−0.923283 + 0.384120i \(0.874505\pi\)
\(3\) −46.5389 + 4.59638i −0.995158 + 0.0982860i
\(4\) −11.8772 − 3.85915i −0.0927909 − 0.0301496i
\(5\) 250.787 − 123.414i 0.897242 − 0.441540i
\(6\) 32.4826 − 553.347i 0.0613933 − 1.04585i
\(7\) −2.39286 − 2.39286i −0.00263678 − 0.00263678i 0.705787 − 0.708424i \(-0.250593\pi\)
−0.708424 + 0.705787i \(0.750593\pi\)
\(8\) −621.573 + 1219.91i −0.429218 + 0.842387i
\(9\) 2144.75 − 427.821i 0.980680 − 0.195620i
\(10\) 979.788 + 3164.76i 0.309836 + 1.00078i
\(11\) −210.579 − 289.837i −0.0477025 − 0.0656568i 0.784499 − 0.620130i \(-0.212920\pi\)
−0.832202 + 0.554473i \(0.812920\pi\)
\(12\) 570.492 + 125.008i 0.0953049 + 0.0208836i
\(13\) −15342.4 + 2429.99i −1.93683 + 0.306763i −0.999168 − 0.0407910i \(-0.987012\pi\)
−0.937660 + 0.347554i \(0.887012\pi\)
\(14\) 32.4496 − 23.5760i 0.00316054 − 0.00229627i
\(15\) −11104.1 + 6896.28i −0.849500 + 0.527589i
\(16\) −14422.0 − 10478.2i −0.880248 − 0.639538i
\(17\) 13948.3 + 7107.04i 0.688575 + 0.350847i 0.763017 − 0.646379i \(-0.223717\pi\)
−0.0744413 + 0.997225i \(0.523717\pi\)
\(18\) 1031.69 + 25901.5i 0.0416961 + 1.04682i
\(19\) −21930.5 + 7125.65i −0.733518 + 0.238334i −0.651874 − 0.758327i \(-0.726017\pi\)
−0.0816438 + 0.996662i \(0.526017\pi\)
\(20\) −3454.92 + 497.997i −0.0965681 + 0.0139194i
\(21\) 122.360 + 100.363i 0.00288317 + 0.00236485i
\(22\) 3783.54 − 1927.81i 0.0757563 − 0.0385997i
\(23\) −36514.8 − 5783.38i −0.625780 − 0.0991139i −0.164513 − 0.986375i \(-0.552605\pi\)
−0.461268 + 0.887261i \(0.652605\pi\)
\(24\) 23320.2 − 59630.2i 0.344345 − 0.880494i
\(25\) 47662.9 − 61901.3i 0.610085 − 0.792336i
\(26\) − 184117.i − 2.05441i
\(27\) −97847.8 + 29768.4i −0.956705 + 0.291060i
\(28\) 19.1861 + 37.6549i 0.000165171 + 0.000324167i
\(29\) 26690.2 − 82143.9i 0.203216 − 0.625435i −0.796566 − 0.604552i \(-0.793352\pi\)
0.999782 − 0.0208833i \(-0.00664785\pi\)
\(30\) −60144.7 − 142781.i −0.406699 − 0.965485i
\(31\) −1992.92 − 6133.58i −0.0120150 − 0.0369784i 0.944869 − 0.327448i \(-0.106189\pi\)
−0.956884 + 0.290469i \(0.906189\pi\)
\(32\) 25487.9 − 25487.9i 0.137502 − 0.137502i
\(33\) 11132.3 + 12520.8i 0.0539247 + 0.0606505i
\(34\) −109064. + 150114.i −0.475888 + 0.655004i
\(35\) −895.409 − 304.784i −0.00353007 − 0.00120158i
\(36\) −27124.7 − 3195.56i −0.0968960 − 0.0114153i
\(37\) −43852.5 − 276874.i −0.142327 − 0.898619i −0.950737 − 0.309999i \(-0.899671\pi\)
0.808410 − 0.588620i \(-0.200329\pi\)
\(38\) −42755.8 − 269949.i −0.126401 − 0.798068i
\(39\) 702849. − 183609.i 1.89730 − 0.495641i
\(40\) −5328.50 + 382647.i −0.0131642 + 0.945341i
\(41\) 14992.3 − 20635.1i 0.0339723 − 0.0467588i −0.791693 − 0.610919i \(-0.790800\pi\)
0.825665 + 0.564160i \(0.190800\pi\)
\(42\) −1401.81 + 1246.36i −0.00291955 + 0.00259579i
\(43\) −132226. + 132226.i −0.253617 + 0.253617i −0.822452 − 0.568835i \(-0.807394\pi\)
0.568835 + 0.822452i \(0.307394\pi\)
\(44\) 1382.57 + 4255.12i 0.00244683 + 0.00753057i
\(45\) 485075. − 371984.i 0.793532 − 0.608528i
\(46\) 135410. − 416750.i 0.205116 − 0.631282i
\(47\) −305041. − 598678.i −0.428565 − 0.841106i −0.999794 − 0.0203148i \(-0.993533\pi\)
0.571229 − 0.820791i \(-0.306467\pi\)
\(48\) 719346. + 421355.i 0.938844 + 0.549925i
\(49\) − 823532.i − 0.999986i
\(50\) 636293. + 672759.i 0.719884 + 0.761139i
\(51\) −681808. − 266642.i −0.719725 − 0.281471i
\(52\) 191603. + 30346.9i 0.188969 + 0.0299297i
\(53\) −1.37672e6 + 701472.i −1.27022 + 0.647210i −0.953524 − 0.301317i \(-0.902574\pi\)
−0.316696 + 0.948527i \(0.602574\pi\)
\(54\) −167067. − 1.20069e6i −0.144382 − 1.03765i
\(55\) −88580.5 − 46698.9i −0.0717908 − 0.0378475i
\(56\) 4406.40 − 1431.73i 0.00335294 − 0.00108944i
\(57\) 987870. − 432421.i 0.706541 − 0.309275i
\(58\) 912159. + 464768.i 0.613864 + 0.312779i
\(59\) −1.18678e6 − 862243.i −0.752292 − 0.546572i 0.144245 − 0.989542i \(-0.453925\pi\)
−0.896536 + 0.442970i \(0.853925\pi\)
\(60\) 158500. − 39056.4i 0.0947324 − 0.0233433i
\(61\) −1.74789e6 + 1.26991e6i −0.985959 + 0.716342i −0.959033 − 0.283296i \(-0.908572\pi\)
−0.0269269 + 0.999637i \(0.508572\pi\)
\(62\) 75500.1 − 11958.0i 0.0402325 − 0.00637220i
\(63\) −6155.79 − 4108.36i −0.00310164 − 0.00207003i
\(64\) −1.09008e6 − 1.50037e6i −0.519793 − 0.715433i
\(65\) −3.54777e6 + 2.50288e6i −1.60235 + 1.13043i
\(66\) −167221. + 107109.i −0.0715957 + 0.0458586i
\(67\) 460041. − 902882.i 0.186868 − 0.366749i −0.778499 − 0.627646i \(-0.784018\pi\)
0.965367 + 0.260897i \(0.0840184\pi\)
\(68\) −138241. − 138241.i −0.0533156 − 0.0533156i
\(69\) 1.72594e6 + 101316.i 0.632492 + 0.0371285i
\(70\) 5228.32 − 9917.30i 0.00182188 − 0.00345581i
\(71\) 3.11520e6 + 1.01219e6i 1.03296 + 0.335627i 0.775958 − 0.630785i \(-0.217267\pi\)
0.256997 + 0.966412i \(0.417267\pi\)
\(72\) −811215. + 2.88231e6i −0.256137 + 0.910075i
\(73\) 573607. − 3.62161e6i 0.172578 − 1.08961i −0.737553 − 0.675290i \(-0.764019\pi\)
0.910130 − 0.414322i \(-0.135981\pi\)
\(74\) 3.32263e6 0.953172
\(75\) −1.93366e6 + 3.09990e6i −0.529255 + 0.848463i
\(76\) 287972. 0.0752495
\(77\) −189.654 + 1197.43i −4.73417e−5 + 0.000298903i
\(78\) 846270. + 8.56859e6i 0.201919 + 2.04446i
\(79\) −5.46749e6 − 1.77650e6i −1.24765 − 0.405387i −0.390573 − 0.920572i \(-0.627723\pi\)
−0.857079 + 0.515185i \(0.827723\pi\)
\(80\) −4.91000e6 − 847911.i −1.07218 − 0.185155i
\(81\) 4.41691e6 − 1.83514e6i 0.923465 − 0.383682i
\(82\) 213774. + 213774.i 0.0428160 + 0.0428160i
\(83\) −4.03654e6 + 7.92215e6i −0.774882 + 1.52079i 0.0769840 + 0.997032i \(0.475471\pi\)
−0.851866 + 0.523760i \(0.824529\pi\)
\(84\) −1065.98 − 1664.23i −0.000196233 − 0.000306363i
\(85\) 4.37517e6 + 60925.8i 0.772731 + 0.0107606i
\(86\) −1.30278e6 − 1.79313e6i −0.220865 − 0.303995i
\(87\) −864568. + 3.94557e6i −0.140761 + 0.642380i
\(88\) 484465. − 76731.7i 0.0757832 − 0.0120029i
\(89\) 3.07550e6 − 2.23448e6i 0.462435 − 0.335979i −0.332051 − 0.943262i \(-0.607740\pi\)
0.794486 + 0.607283i \(0.207740\pi\)
\(90\) 3.45535e6 + 6.36842e6i 0.499624 + 0.920838i
\(91\) 42526.8 + 30897.5i 0.00591585 + 0.00429812i
\(92\) 411376. + 209607.i 0.0550785 + 0.0280639i
\(93\) 120941. + 276290.i 0.0155913 + 0.0356184i
\(94\) 7.57424e6 − 2.46102e6i 0.940570 − 0.305610i
\(95\) −4.62047e6 + 4.49355e6i −0.552909 + 0.537721i
\(96\) −1.06903e6 + 1.30333e6i −0.123322 + 0.150351i
\(97\) 1.77246e6 − 903112.i 0.197185 − 0.100471i −0.352610 − 0.935770i \(-0.614706\pi\)
0.549795 + 0.835299i \(0.314706\pi\)
\(98\) 9.64096e6 + 1.52698e6i 1.03474 + 0.163886i
\(99\) −575638. − 531537.i −0.0596247 − 0.0550568i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.8.l.a.2.19 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.50 yes 544
25.13 odd 20 inner 75.8.l.a.38.50 yes 544
75.38 even 20 inner 75.8.l.a.38.19 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.19 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.50 yes 544 3.2 odd 2 inner
75.8.l.a.38.19 yes 544 75.38 even 20 inner
75.8.l.a.38.50 yes 544 25.13 odd 20 inner